UNIT 4 Part 2 4.6 Name:___________________________Complete the Square, Vertex Form, Domain and Range

Quadratic Functions and their Graphs:General Form:

Vertex Form: , where is the vertex, x = h is the axis of symmetry, and a determines the shapeand direction in which it opens.

Equation of the Axis of Symmetry:

Transformations of Quadratic Functions: In unit 2, you learnedhow different transformations affect the graph of parent functions. Summary of transformations : horizontal translation – h to the right, or h to the left, vertical translation – k units up if positive, k units down if negative reflection over x-axis – if a is positive opens up, if a is negative opens down dilations – if a is a whole number, vertical stretch, if a is a fraction (smaller than 1) vertical compression

To Graph From Vertex Form:1. Plot the vertex point, .2. Use a to determine: opens up or down, fat or skinny3. Choose x values to plot 2 other points.

Example 1: Graph the equation

1. Plot the vertex point, .2. a = -1, flips overs the x-axis3. Choose another x-value to plot an addition pt. on the graph.

To Write The Equation in Vertex Form from General Form:1. Find the equation of the axis of symmetry.2. Plug this x-value into the original equation to find y, which will be your k.3. Use the a from the original equation.

Example 2: Write in vertex form.

1.

2.

VERTEX

3.

Some quadratic functions do not factor. These quadratic functions can be solved using a method calledcompleting the square. Completing the square is a process by which you can force a quadratic expression to factor.

To Write in vertex form by Completing the square, then solve:1. Put any constants on the opposite side of the equation2. If the leading term has a coefficient (other than 1), factor out that coefficient, and divide both sides by it3. Take half the linear term.4. Square the number from step 3.5. Add it to both sides of the equation.6. Write the equation in factored form on the left hand side7. Then solve using the square root method.

Example 3: Solve the following quadratic equation by completing the square.= 0 coeff. of linear term is – 6, half of that is – 3 , square it

then add it to both sides of the equation

write the equation in factored form Note: half the linear term will always be your factor Vertex form:

To SOLVE: Take the square root of both sides to get rid of the square

solve for x and write the resulting 2 equations, solve for x in each equation x = 6 and x = 0 add 3 to both sides, on both equations

Example 4: Solve An equation with a constant add 16 to both sides to isolate the squared and linear terms add half the linear coeff. squared , to both sides

write your factor, simplify the right side

take the square root of both sides

and write the 2 equations, solve for x x = 2 and x = -8 subtract 3 from both sides of the equations

Example 5: Solve when the leading term has a coeff. other than 1. add 5 to get the squared and linear term by themselves

factor out the leading coefficient

divide both sides by 3 to get the squared term by itself

add half the coeff. of the linear term squared to both sides

get a common denominator to combine on the right side

combine like terms

take the square root of both sides

and write the 2 equations, solve for x x = 2.63 and x = - 0.63 add 1 to both sides

Unit 4 Part 2, 4.6 Worksheet Name:_______________________

Graph the following. No Calculators. Then state the domain and range in interval notation. Label the vertex point.

1. 2. 3.

4. 5. 6.

Write each function in vertex form. Then solve.7. 9. 11. 8. 10. 12.

Solve each quadratic equation by completing the square.13. 15. 17. 14. 16. 18.